A Numerical Algorithm for the Analysis of the Thermal Stress-Stain State of a Rod

A universal algorithm for solving applied tasks of thermo elastic state of a partially heat insulated rod of limited length, in the presence of axial force, temperature, heat flow and heat exchange, was developed. The corresponding numerical calculation of efforts of the rod partially thermally insulated and clamped by two ends, in the presence of heat flow was elaborated. The numerical algorithm allowed calculate the lengthening value of the partially thermally insulated rod or the compressive stress and strain in the rods of limited length with the heat flow, the heat exchange, heat insulation and the axial tensile force.


Introduction
Construction of numerical algorithms and complex of programs, as well as the solution of the task of the steady thermal elastic state of a partially isolated heat rod while the simultaneous effects of axial force, temperature and heat flow represent the appropriate mathematical and computational difficulties. In view of the importance of ensuring stable work of construction elements, while the simultaneous thermal and mechanical action, we are really convinced that the development of universal computing algorithms of established processes, of the field temperature distribution, of elastic displacements and thermo elastic condition of construction elements with mechanical forces, heat flow, partial insulation and heat exchange is the actual problem of modern numerical methods.
With simultaneous consideration of the above factors, the analytical solution of the steady thermo elastic compression and stretching of the rods becomes very complicated. In this connection, the need naturally arises to develop appropriate universal computational algorithms oriented to modern computer facilities and object-oriented high-level programming languages that allow one to investigate the thermo elastic state of partially insulated one-dimensional structural elements, in the presence of heat exchange with the environment, the temperature field, thermal and axial tensile forces.
On the basis of the energy principles and finite element method with quadratic elements, the universal numerical algorithm for solving applied tasks of thermo elastic state of a partially heat insulated rod of limited length, in the presence of axial force, temperature, heat flow and heat exchange, was developed.
The field regularities of distribution of the temperature, displacement, deformation and strain along the length of the rod, depending on the area of the lateral heat-isolated surface, on the applied temperature, heat flow, heat exchange and heat exchange coefficients with the surrounding parts of the lateral surface, were revealed. Earlier the authors made the numerical study of lengthening of partially heat-isolated rod in the presence of a heat sources and tensile strength is given. The numerical study with the different source data was conducted [1].

Calculation Scheme of the Task
To do it, the rod L with length will be to discrete quadratic finite elements with three nodes.
Then, for the final elements in the heat isolated lateral surface parts, the view o the functional, expressing conservation and change of the total thermal energy is as follows For the finite element heat isolated by the lateral surface, the functional type is as follows On the lateral surface 2 1 x x x ≤ ≤ of the rod, the heat exchange with the environment takes place.
On the lateral surface 4 3 x x x ≤ ≤ of the rod, the heat exchange with the environment takes place as well.
Then, for the rod in general, the expression of the corresponding functional is as follows The influence of temperature and tensile strength on the lengthening of the rod was analyzed [1].
Moreover, the lengthening of considered rod is affected the by heat transfer coefficient between the rod material and the cross-sectional area of the clamped rod ends [2].

The Payment Scheme
The numerical study of thermal strain deformed of partially heat isolated and clamped by two rod ends of limited length in the presence of different heat sources was carried out [3].
The field of temperature distribution along the length of the rod, the law of distribution of elastic displacements, strain and stress components along the length of the rod was defined. For this, the length of the rod is divided into equal n -parts. Then the length of one side will be n L l / = . The number of nodes of the discrete elements will be 1 2 + n . The field of distribution of temperature and elastic displacement within each element is approximated by a quadratic curve passing through three points.
For the first element the form of the functional that expresses the full heat energy is as follows And for all other elements of the portion 1 0 x x ≤ ≤ of the rod, the analogous functional form will be as follows 2  2   2  2  2   7  16  2  3  3  3  16  16  7  2  2  3  3 The last n element was considered. The lateral surface of this element was thermally insulated. But through cross sectional area of the right end, which corresponds to the section L x = , a heat exchange with the environment takes place. Then this element is characterized by full expression of functional thermal energy has the following form Then we present the second part of the considered problem. For each quadratic finite element with the length 1 l the expression of the potential energy of elastic deformations, taking into account the available steady temperature field, is Within this element, the expression of the elastic component of deformations is as follows After a little simplification for i-quadratic finite element, the expression of the potential energy of elastic deformation (11) has the following form Then, the form of the functional expressing the potential energy of the considered rod clamped at both ends will be as follows Next we present the minimization of the total potential energy of elastic deformation of the considered rod by the nodal values of displacements We determine the value of the elastic displacements of the rod sections that correspond to the element nodes. After that, we propose the algorithms for computing the components of the elastic and thermal components of deformations and strains for half of each finite element [4].

The Analysis of the Effect
After testing of the developed computational algorithm, the influence of local temperature and heat exchange on the thermal-strained and deformed state of the test rod is analyzed [5]. To do this, the values of the compressive force and the true stress on different rod parts are calculated      On the basis of the developed mathematical model, corresponding to the computational algorithm, the set of applications for solving the class of problems in order to determine the temperature distribution field, displacements, deformations, strains was developed. This complex will make it possible to calculate the lengthening value of the partially thermally insulated rod or the compressive stress and strain in the rods of limited length with the heat flow, the heat exchange, heat insulation and the axial tensile force.